#416930

order	512
nilpotency class	3
isoclinism	isoclinic class of #395552
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#394
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1052 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 40 points:

(1,2)(3,7)(4,5)(6,8)(9,10,13,20,14,18,15,19)(11,21,23,16,24,35,28,31)(12,17,29,34,27,33,26,36)(22,32,25,30,39,37,38,40);(2,5)(6,7)(9,11,15,28,14,24,13,23)(10,16,20,35,18,31,19,21)(12,25,29,39,27,38,26,22)(17,32,36,40,33,37,34,30);(9,12)(10,17)(11,22)(13,26)(14,27)(15,29)(16,30)(18,33)(19,34)(20,36)(21,37)(23,38)(24,39)(25,28)(31,40)(32,35);(1,3,4,8)(2,6,5,7)(10,18)(16,31)(17,33)(19,20)(21,35)(30,40)(32,37)(34,36)

Polycyclic presentation:

2^9|a^2=fg,b^2=eh,d^2=hi,e^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,g^a=gi,c^b=cg,f^b=fi,g^b=gi,g^c=gi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 324126);